2009 PIMS-CSC Seminar - 02

Superlattice patterns and quasipatterns, while well-studied in waves
on the surface of vertically vibrated viscous fluids (Faraday waves),
have found little attention in forced oscillatory systems. We study
such patterns, comprised of 4 or more Fourier modes at different
orientation, by applying multi-frequency forcing to systems undergoing
a Hopf bifurcation to spatially homogeneous oscillations. For weak
forcing composed of 3 frequencies near the 1:2- and 1:3-resonance,
such systems can be described by a suitably extended complex
Ginzburg-Landau equation with time periodic coefficients. Using
Floquet theory and weakly nonlinear analysis we obtain the amplitude
equations for simple patterns (comprised of 1, 2, or 3 modes) and
superlattice patterns. By judicious choice of the forcing function we
stabilize these patterns via spatiotemporal resonance and find stable
subharmonic 4- and 5-mode patterns. We confirm our analysis through
numerical simulation.